There was no way for the Jedi to save the Republic while being part of the same Republic that was failing. The moment they integrated themselves into the politics and bureaucracy all those years ago was the moment the seeds for their failure were planted.
And even if they could "save" the Republic, by the time of TPM the Republic isn't even worth saving. It's a government where corporations are awarded senators and are allowed to blockade sovereign planets to protest taxation. That would be like if Amazon encircled Rhode Island and didn't let anyone in or out because they didn't like digital platforms having to pay sales tax. And in this situation, it's entirely legal. Sure, when the Trade Federation invaded Naboo they crossed a line, and that was the whole reason they wanted Amidala to sign a treaty legalizing it (whether or not the Republic would have recognized a treaty signed under duress as binding is another matter, though I wouldn't doubt it), but the fact that they could even go all the way up that line, and no one thought anything wrong with it shows something. The fact that Chancellor Valorum had to send two Jedi in secret and circumvent the senate in order to get anything done shows how useless and corrupt as an institution the Republic had become.
There's a reason the Separatists wanted out of the Republic. Dooku didn't make them leave, he just organized the uprising. But it was something they were already feeling to begin with. Why would they want to be part of a government that exploits their resources and taxes them, but gives them no protection or real representation in return. Heck, a lot of planets even had to rely on the Trade Federation for protection, since they at least had an interest in keeping pirates at bay. Their grievances were real. Same with Dooku. Him leaving the Jedi Order was because he lost faith in the Republic, and rightfully so. No matter what the Separatists and Dooku did later on, their reasons for leaving were legitimate.
And anyone who thinks "The Jedi were fools for using an army they just discovered!" doesn't realize that it's not the Jedi's army. The Clone Army belongs to the Republic. The Senate votes Palpatine emergency powers, which he uses to create a Grand Army of the Republic. At this point the Republic already know the Separatists have an army and are preparing to put a gun to the Republic's head in order to forcefully allow them to succeed, which is why they grant Palpatine that power to begin with. So once he orders the clones to be used, they're going off to fight the Separatists, whether or not the Jedi go along. And given that the Jedi are the historical protectors of the Republic and have fought in prior wars, there's not much reason for them not to go and fight...and especially since the enemy is former Jedi himself. But again, whether or not they go, the Clone Wars are happening, even if they sit on the sidelines. And even if somehow them not fighting alongside the clones prevents everything, in the worst case scenario the Republic simply falls to the Separatists and is replaced externally instead of internally. The biggest difference in that scenario is that Palpatine simply becomes the ruler behind the scenes instead of the face of this new Empire, but in practice the outcome isn't much different.
In the end the Republic couldn't be saved. Dooku was right about that. And it probably wasn't even worth saving, either.
Weren't Jedi convinced Dooku was Maul's master? They didn't expect another Sith.
Yeah, in the "Darth Maul: Son of Dathomir" series based on the unfinished TCW episodes, the Jedi eventually come to the conclusion that Dooku trained Maul, and Maul is his apprentice, which they inform Palpatine of (who must find it superbly amusing). It's basically a matter of them misinterpreting events (they find Maul and Dooku working together, and don't realize it's an temporary alliance of convince), as well as trying to make all the pieces fit together, even if they have to force them, in order for them not to have to admit to worse truths.
